Delay and Doppler spreads in underwater acoustic particlevelocity channels

Huaihai Guoa) and Ali AbdiCenter for Wireless Communication and Signal Processing Research, Department of Electrical and ComputerEngineering, New Jersey Institute of Technology, Newark, New Jersey 07102

Aijun Song and Mohsen BadieyPhysical Ocean Science and Engineering, College of Earth, Ocean, and Environment, University of Delaware,Newark, Delaware 19716

(Received 24 March 2010; revised 9 December 2010; accepted 10 December 2010)

Signal processing and communication in acoustic particle velocity channels using vector sensors

are of interest in the underwater medium. Due to the presence of multiple propagation paths, a

mobile receiver collects the signal with different delays and Doppler shifts. This introduces certain

delay and Doppler spreads in particle velocity channels. In this paper, these channel spreads are

characterized using the zero-crossing rates of channel responses in frequency and time domain.

Useful expressions for delay and Doppler spreads are derived in terms of the key channel parameters

mean angle of arrival and angle spread. These results are needed for design and performance

prediction of systems that utilize underwater acoustic particle velocity and pressure channels.VC 2011 Acoustical Society of America. [DOI: 10.1121/1.3533721]

PACS number(s): 43.60.Cg, 43.60.Dh [NPC] Pages: 2015–2025

I. INTRODUCTION

Data communication is of interest in numerous naval and

civilian applications. Examples include communication

among autonomous underwater vehicles (AUVs) for collabo-

rative operations, harbor security systems, tactical surveillance

applications, oceanographic data retrieval from underwater

sensors over geographically large areas, offshore oil and gas

explorations, etc. After the first generation of analog modems,

second generation digital modems in 80’s used noncoherent

techniques such as frequency shift keying and differentially

coherent schemes like differential phase shift keying

(DPSK).1 Due to the need for higher spectral efficiencies,

coherent systems with phase shift keying and quadrature am-

plitude modulation were developed in 90’s.2,3 Spatial diversity

with arrays of hydrophones and different types of equalization,

beamforming, coding, channel estimation, and tracking are

also used for underwater communication.1 Underwater multi-

ple-input multiple-output (MIMO) systems using spatially

separated pressure sensors are also recently investigated.4–7

A vector sensor can measure non-scalar components of

the acoustic field such as the particle velocity, which cannot

be sensed by a single scalar (pressure) sensor. Development

of vector sensors dates back to 30’s.8 In the past few decades,

a large volume of research has been conducted on theory, per-

formance evaluation, and design of vector sensors. They have

been mainly used for target localization and sound navigation

and ranging (SONAR) applications. Examples include accu-

rate beamforming and azimuth/elevation estimation of a

source, avoiding the left–right ambiguity of linear towed

arrays of scalar sensors, significant acoustic noise reduction

due to the highly directive beam pattern, etc.9–12

Vector sensors have recently been proposed13,14 and

then used15,16 for underwater acoustic communication.

Characterization of particle velocity channels and their

impact on vector sensor communication systems perform-

ance is therefore of interest. In multipath channels such as

shallow waters, a vector sensor receives the signal through

several paths and each path has a different delay (travel

time). Motion of the transmitter or receiver in a multipath

channel introduces different Doppler shifts as well. The

largest differences between the path delays and between

the Doppler shifts are called delay spread and Doppler

spread, respectively.17 The harsh multipath, with delay

spreads up to hundreds of symbols for high data rates, and

temporal variations of the underwater acoustic channels,

with Doppler spreads up to several tens of hertz, are

major issues in underwater acoustic communication.1

Some typical values for delay spread18,19 and Doppler

spread20,21 of the underwater acoustic channel are 5–15

ms and 6–30 Hz, respectively. Knowledge of delay and

Doppler spreads in acoustic particle velocity channels is

important for efficient design of underwater vector sensor

communication systems. Characterization of delay and

Doppler spreads in terms of the physical parameters of the

propagation environment is needed for system perform-

ance prediction as well.

In general, the zero-crossing rate (ZCR) of a random

process carries useful information that can be used for vari-

ous purposes. Examples include signal detection,22 estima-

tion of the density of scatterers,23 calculating sonar false

alarm probability,24 characterization of acoustic emission

signals,25 speech recognition,26 etc.

It is also well known that delay and Doppler spreads are

proportional to the ZCRs of the channel in frequency27,28

and time29,30 domains, respectively. In Fig. 1 the physical

meaning of frequency-domain ZCR and its connection with

a)Author to whom correspondence should be addressed: Electronic mail:

hg45@njit.edu

J. Acoust. Soc. Am. 129 (4), April 2011 VC 2011 Acoustical Society of America 20150001-4966/2011/129(4)/2015/11/$30.00

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delay spread are shown. Let the Fourier pair P(f) and p(s)

represent the complex channel transfer function and impulse

response, respectively, where f and s are frequency and

time delay in hertz and seconds, respectively. As shown in

Fig. 1(a), Re{P(f)} slowly varies with f, where Re{.} gives

the real part. This slow variation results in a small number of

times that the channel transfer function crosses the zero

level, i.e., low ZCR. In this figure there are nine zero cross-

ings over 10 Hz, which gives a ZCR of 0.9. The magnitude

of the channel impulse response, |p(s)|, is shown in Fig. 1(b),

which has a relatively small delay spread. This is because of

the well-known inverse relationship between frequency-axis

and time-axis scalings, i.e., waveform expansion in one do-

main results in compression in the other domain. On the

other hand, when the ZCR of the channel transfer function is

higher, as shown in Fig. 1(c), one can say the channel trans-

fer function is compressed in the f domain. This results in

the expansion of the channel impulse response in the s do-

main shown in Fig. 1(d), which now has a larger delay

spread. The connection between time-domain ZCR and

Doppler spread can be similarly explained.

To calculate the frequency- and time-domain ZCRs, one

needs to obtain the second derivative of the corresponding

frequency and temporal channel correlations, respectively,

as explained in Sec. IV. Therefore, to calculate delay and

Doppler spreads in particle velocity channels, expressions

for frequency and temporal correlations of such channels

should be derived first.

In what follows, basic formulas and definitions for particle

velocity channels are provided in Sec. II A statistical model

for particle velocity channels sensed by a moving vector sensor

array is developed in Sec. III and channel correlation functions

are derived as well. Using the frequency and temporal correla-

tion functions, frequency- and time-domain ZCRs are calcu-

lated in Sec. IV for both pressure and particle velocity

channels. Numerical results and concluding remarks are pro-

vided in Secs. V and VI, respectively.

II. DEFINITIONS OF PARTICLE VELOCITY CHANNELS

As an underwater acoustic communication receiver, we

consider the vector sensor array shown in Fig. 2, in the two-

dimensional y–z (range-depth) plane. There is one pressure

sensor transmitter in the far field, called Tx, shown by a

black circle. We have three receive vector sensors Rx, Rx1,

and Rx2, represented by black squares, with the center of

array located at y¼ 0 and z¼D. Each vector sensor meas-

ures the pressure, as well as the y and z components of the

acoustic particle velocity, all in a single co-located point.

This means there are three pressure channels p, p1, and p2,

as well as six pressure-equivalent velocity channels py, pz,

py1, pz

1 , py2 , and pz

2, all measured in Pascal (Newton/m2). In

Fig. 2 pressure channels are represented by solid straight

lines, whereas the y-velocity channels and the z-velocity

channels are represented by dashed curved lines and dotted

curved lines, respectively. The particle velocity channels vy,

vz, vy1, vz

1, vy2, and vz

2 are defined as

vy ¼ � 1

jq0x0

@p

@y; vz ¼ � 1

jq0x0

@p

@z;

vy1 ¼ �

1

jq0x0

@p1

@y; vz

1 ¼ �1

jq0x0

@p1

@z;

vy2 ¼ �

1

jq0x0

@p2

@y; vz

2 ¼ �1

jq0x0

@p2

@z: (1)

In the above equations q0 is the density of the fluid in kg/m3,

j2¼�1 and x0¼ 2pf0 is the frequency in rad/s. By multiply-

ing the velocity channels in Eq. (1) with �q0c, the negative

of the acoustic impedance of the fluid, where c is the sound

speed in m/s, we obtain the pressure-equivalent velocity

FIG. 1. (Color online) Graphical representation of the connection between

the ZCR of a channel transfer function and the delay spread of the corre-

sponding channel impulse response: (a) A channel transfer function with

low ZCR, (b) the corresponding channel impulse response with small delay

spread, (c) a channel transfer function with high ZCR, and (d) the corre-

sponding channel impulse response with large delay spread.

FIG. 2. A system with one pressure transmitter and three vector sensor

receivers. Each vector sensor measures the pressure, and the y and z compo-

nents of the acoustic particle velocity.

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channels py, pz, py1, pz

1, py2, and pz

2. Note that py¼�q0cvy,

pz¼�q0cvz, py1 ¼ �q0cvy

1 , pz1 ¼ �q0cvz

1 , py2 ¼ �q0cvy

2, and

pz2 ¼ �q0cvz

2. With k as the wavelength in m and k = 2p/k¼x0 /c as the wavenumber in rad/m, finally we obtain,

py ¼ 1

jk

@p

@y; pz ¼ 1

jk

@p

@z;

py1 ¼

1

jk

@p1

@y; pz

1 ¼1

jk

@p1

@z;

py2 ¼

1

jk

@p2

@y; pz

2 ¼1

jk

@p2

@z: (2)

The received rays at the vector sensor array are shown

in Fig. 3 where D0 is the water depth. Vector sensor 1 is

located at y¼ Ly=2 and z¼D � (Lz=2), vector sensor 2

is at y¼� Ly=2 and z¼Dþ (Lz=2) and vector sensor Rx is

located at y¼ 0 and z¼D. Here, Ly and Lz are the projec-

tions of the array length L at y and z axis, respectively,

such that L ¼ ðL2y þ L2

z Þ1=2

. All the angles are measured

with respect to the positive direction of y, counterclock-

wise. We model the rough sea bottom and its surface as

collections of Nb and Ns scatterers, respectively, such that

Nb� 1 and Ns� 1. In this paper, the small letters b and srefer to the bottom and surface, respectively. In Fig. 3, for

example, the i-th bottom scatterer is represented by Obi ,

i¼ 1, 2, …, Nb, whereas Osm denotes the m-th surface scat-

terer, m¼ 1, 2, …, Ns. Rays scattered from the bottom and

the surface toward the vector sensors are shown by solid

lines. The rays scattered from Obi hit Rx1 and Rx2 at the

angle of arrivals (AOAs) cbi;1 and cb

i;2 , respectively. The

traveled distances are labeled by nbi;1 and nb

i;2 , respectively.

Similarly, the scattered rays from Osm impinge Rx1 and Rx2

at the AOAs csm;1 and cs

m;2 , respectively, with nsm;1 and

nsm;2 as the traveled distances shown in Fig. 3. The vector

sensor receivers move at the speed u, in the direction speci-

fied by u in Fig. 3.

III. CHANNEL CORRELATION FUNCTIONS

In a multipath channel, delay spread and Doppler spread

are key channel characteristics for system design. As dis-

cussed previously, delay and Doppler spreads are represented

by frequency- and time-domain ZCRs of the channel, respec-

tively. Furthermore, frequency- and time-domain ZCRs are

related to the second derivative of frequency and temporal

channel correlations, respectively. To obtain these channel

correlation functions, in what follows we consider a statistical

framework that leads us to frequency-time-space correlation

functions for particle velocity channels. By taking proper

derivatives of these correlation functions, ZCRs of interest

will be obtained.

Superposition of plane waves at the mobile sensors Rx1

and Rx2 results in the following time-varying transfer func-

tions P1(f, t) and P2(f, t) for the pressure channels,

P1ðf ; tÞ ¼ðKb=NbÞ1=2XNb

i¼1

abi expðjwb

i Þ�

� expðjk½y cosðcbi;1Þ þ z sinðcb

i;1Þ�Þ� expð�j2pf sb

i;1Þ

� expðj2pfM cosðcbi;1 � uÞtÞ

o���y¼Ly=2;z¼D�Lz=2

þ ðð1� KbÞ=NsÞ1=2XNs

m¼1

asm expðjws

m�

� expðjk ½y cosðcsm;1Þ þ z sinðcs

m;1Þ�Þ� expð�j2pf ss

m;1Þ

� expðj2pfM cosðcsm;1 � uÞtÞ

o���y¼Ly=2;z¼D�Lz=2

; (3)

P2ðf ; tÞ ¼ðKb=NbÞ1=2XNb

i¼1

abi expðjwb

i Þ�

� expðjk½y cosðcbi;2Þ þ z sinðcb

i;2Þ�Þ� expð�j2pf sb

i;2Þ

� expðj2pfM cosðcbi;2 � uÞtÞ

o���y¼�Ly=2;z¼DþLz=2

þ ðð1� KbÞ=NsÞ1=2XNs

m¼1

nas

m expðjwsmÞ

� expðjk½y cosðcsm;2Þ þ z sinðcs

m;2Þ�Þ� expð�j2pf ss

m;2Þ

� expðj2pfM cosðcsm;2 � uÞtÞ

o���y¼�Ly=2;z¼DþLz=2

: (4)

Note that P1(f, t) in Eq. (3) is the sum of two summations,

one multiplied by (Kb=Nb)1/2 and the other one by

((1� Kb)=Nb)1/2. The first summation has Nb terms, and

each term is the product of abi and four complex exponen-

tials expðjwbi Þ, expðjk ½y cosðcb

i;1Þþz sinðcbi;1Þ�Þ, expð�j2pf sb

i;1Þand expðj2pfM cosðcb

i;1 � uÞtÞ. The second summation in Eq.

(3) has Ns terms, and each term is the product of asm and

FIG. 3. Received rays at the three vector sensors in a multipath shallow

water channel.

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four other complex exponentials. Similar explanations

hold for P2(f, t) and its two summations in Eq. (4). In

Eqs. (3) and (4), abi > 0 and as

m > 0 represent the ampli-

tudes of the rays scattered from Obi and Os

m, respectively,

whereas wbi 2 ½0; 2pÞ and ws

m 2 ½0; 2pÞ stand for the asso-

ciated phases. The delays s in Eqs. (3) and (4) represent

the travel times from the bottom and surface scatterers to

the two vector sensors. For example, sbi;1 denotes the

travel time from Obi to Rx1. Moreover, the AOAs c in

Eqs. (3) and (4) stand for the propagation directions of

waves scattered from the bottom and surface toward the

receivers. For instance, csm;2 is the AOA of a ray at Rx2

which is coming from Osm . The complex exponential of

the form exp(jk[y cos (c) þ z sin (c)]) in Eqs. (3) and (4)

represent a plane wave with the AOA c at the point (y, z),

exp(�j2pfs) specifies that the plane wave’s travel time is

s,31 and exp(j2pfMcos (c � u)t) corresponds to its Doppler

shift introduced by the motion of the receiver.32,33 Here

fM¼ u/k is the maximum Doppler shift. The terms (Nb)�1/

2 and (Ns)�1/2 are included in Eqs. (3) and (4) to scale the

power. Moreover, 0�Kb� 1 represents the amount of the

contribution of the bottom scatterers, whereas the contri-

bution of the surface scatterers is given by 1 � Kb.

Based on the definition of the correlation function

CP2P1(Kf, Kt, Lz, Ly)¼E½P2ðf þ Df ; tþ DtÞP�1ðf ; tÞ� and

the channel representations in Eqs. (3) and (4), an integral

expression can be similarly31,34 obtained for the frequency-

time-space pressure channel correlation,

CP2P1ðDf ;Dt; Lz; LyÞ ¼ Kb

ðpcb¼0

wbðcbÞ exp½�jkLyðcosðcb2Þ þ cosðcb

1ÞÞ=2� exp½jkDðsinðcb2Þ � sinðcb

1ÞÞ�� exp½jkLzðsinðcb

2Þ þ sinðcb1ÞÞ=2� exp½j2pf ðsb

1� sb

2Þ�

� exp½�j2pDf sb2� exp½j2pfM cosðcb � uÞDt�

8>><>>:

9>>=>>;dcb

þ ð1� KbÞð2p

cs¼p

wsðcsÞ exp½�jkLyðcosðcs2Þ þ cosðcs

1ÞÞ=2� exp½jkDðsinðcs2Þ � sinðcs

1ÞÞ�� exp½jkLzðsinðcs

2Þ þ sinðcs1ÞÞ=2� exp½j2pf ðss

1� ss

2Þ�

� exp½�j2pDf ss2� exp½j2pfM cosðcs � uÞDt�

8><>:

9>=>;dcs: (5)

Each integrand in Eq. (5) is the product of a w function

defined later, and six complex exponentials. Note that cb

and cs are the AOAs of rays coming from bottom and sur-

face toward the array center, respectively. The terms

cosðcbqÞ, cosðcs

qÞ, sinðcbqÞ and sinðcs

qÞ, q = 1, 2, represent the

corresponding cosine and sine of bottom and surface

AOAs for the vector sensors Rx1 and Rx2, respectively.

Moreover, sbq and ss

q , q = 1, 2 are travel times from the

bottom and surface scatterers to Rx1 and Rx2, respec-

tively. As shown in Appendix A, all these parameters can

be written in terms of D, D0, cb, cs, Lz, and Ly. Finally,

wb(cb) and ws(cs) in Eq. (5) are the probability density

functions (PDFs) of the AOAs of the waves coming from

the sea bottom and surface, respectively, such that 0 < cb

< p and p < cs <2p.

Similarly,31 to obtain pressure–velocity and velocity–

velocity channel correlations, one need to take proper deriv-

atives of the pressure channel correlation in Eq. (5), as sum-

marized below:

CP2Pz1ðDf ;Dt; Lz; LyÞ ¼ E½P2ðf þ Df ; tþ DtÞfPz

1ðf ; tÞg��

¼ ðjkÞ�1@CP2P1ðDf ;Dt; Lz; LyÞ=@Lz; (6)

CP2Py1ðDf ;Dt; Lz; LyÞ ¼ E½P2ðf þ Df ; tþ DtÞfPy

1ðf ; tÞg��

¼ ðjkÞ�1@CP2P1ðDf ;Dt; Lz; LyÞ=@Ly; (7)

CPz2Pz

1ðDf ;Dt; Lz; LyÞ ¼ E½Pz

2ðf þ Df ; tþ DtÞfPz1ðf ; tÞg

��¼ �k�2@2CP2P1

ðDf ;Dt; Lz; LyÞ=@L2z ;

(8)

CPy2Py

1ðDf ;Dt;Lz;LyÞ ¼ E½Py

2ðf þ Df ; tþDtÞfPy1ðf ; tÞg

��¼ �k�2@2CP2P1

ðDf ;Dt;Lz;LyÞ=@L2y ;

(9)

CPz2Py

1ðDf ;Dt; Lz; LyÞ ¼ E½Pz

2ðf þ Df ; tþ DtÞfPy1ðf ; tÞg

��¼ �k�2@2CP2P1

ðDf ;Dt; Lz; LyÞ; =@Lz@Ly:

(10)

In Eqs. (6)–(10), the time-varying transfer functions for the

pressure-equivalent velocity channels at the q-th vector sen-

sor, q¼ 1, 2, are defined as Pzqðf ; tÞ ¼ ðjkÞ

�1@Pqðf ; tÞ=@zand Py

qðf ; tÞ ¼ ðjkÞ�1@Pqðf ; tÞ=@y .

Now, we focus on the vector sensor receiver Rx, to cal-

culate the delay and Doppler spreads of particle velocity

channels. With Lz¼Ly¼ 0 in Eq. (5), it is straight forward to

obtain the frequency-time autocorrelation of P(f, t), the time-

varying transfer function of the pressure channel, at Rx,

CPPðDf ;DtÞ ¼ E½Pðf þ Df ; tþ DtÞP�ðf ; tÞ�¼ CP2P1

ðDf ;Dt; 0; 0Þ¼ KbEcb exp½�j2pDfTb=sinðcbÞ�

�� exp½j2pfM cosðcb � uÞDt�

�þ ð1� KbÞEcs exp½�j2pDfTs=sinðcsÞ�½� exp½j2pfM cosðcs � uÞDt��: (11)

Here Tb¼ (D0 � D)=c, the vertical travel time from bottom

to Rx and Ts¼D=c, the vertical travel time from surface to

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Rx. Equation (11) is obtained because when Lz¼ Ly¼ 0,

Eqs. (A1)–(A9) in Appendix A result in cb2 ¼ cb

1, sb2 ¼ sb

1,

sb2 ¼ Tb=sinðcbÞ, cs

2 ¼ cs1 , ss

2 ¼ ss1 , and ss

2 ¼ Ts=sinðcsÞ. Let

us define the time-varying transfer functions for the pres-

sure-equivalent velocity channels at the vector sensor Rx as

Pz(f, t)¼ (jk)�1 @P(f, t)=@z and Py(f, t)¼ (jk)�1@P(f, t)=@y.

Then using Eqs. (8) and (9), the second derivative of Eq. (5)

at Lz¼Ly¼ 0 provide the following frequency-time autocor-

relations of Pz(f, t) and Py(f, t) at Rx, respectively.

CPzPzðDf ;DtÞ ¼ E½Pzðf þ Df ; tþ DtÞ Pzðf ; tÞf g��¼ CPz

2Pz

1ðDf ;Dt; 0; 0Þ

¼ KbEcb sin2ðcbÞ exp½�j2pDfTb=sinðcbÞ��

� exp½j2pfM cosðcb � uÞDt��

þð1� KbÞEcs sin2ðcsÞ�

� exp½�j2pDfTs=sinðcsÞ�� exp½j2pfM cosðcs � uÞDt�

�; (12)

CPyPyðDf ;DtÞ ¼ E½Pyðf þ Df ; tþ DtÞ Pyðf ; tÞf g��¼ CPy

2Py

1ðDf ;Dt; 0; 0Þ

¼KbEcb cos2ðcbÞ exp½�j2pDfTb=sinðcbÞ��

� exp½j2pfM cosðcb � uÞDt��

þ ð1� KbÞEcs cos2ðcsÞ�

� exp½�j2pDfTs=sinðcsÞ�� exp½j2pfM cosðcs � uÞDt�

�: (13)

Before concluding this section, we derived the correla-

tion between the real parts of P1(f, t) and P2(f, t) in Eqs. (3)

and (4). This will be needed to compute the ZCR of real

channel functions in the next section. If W is complex, then

Re{W}¼ (W þ W*)=2. This results in

E Re P2ðf þ Df ; tþ DtÞf gRe P1ðf ; tÞf g½ �

¼ 1

2Re CP2P1

ðDf ;Dt; Lz; LyÞ� �

þ 1

2Re E P2ðfþDf ; tþ DtÞP1ðf ; tÞ½ �g; (14)f

where CP2P1ðDf ;Dt; Lz; LyÞ is given in Eq. (5). Based on the

statistical properties of phases wbi and ws

m, independent and

uniformly distributed over 2p radians, it can be verified that

the second term on the right hand side of Eq. (14) is zero.

This yields

E Re P2ðf þ Df ; tþ DtÞf gRe P1ðf ; tÞf g½ �

¼ 1

2Re CP2P1

ðDf ;Dt; Lz; LyÞ� �

: (15)

Similarly, based on the definitions of Pzqðf ; tÞ and Py

qðf ; tÞ,q¼ 1, 2, provided after Eq. (10), we obtain,

E Re Pz2ðf þ Df ; tþ DtÞ

� �Re Pz

1ðf ; tÞ� �� �

¼ 1

2Re CPz

2Pz

1ðDf ;Dt; Lz; LyÞ

n o; (16)

E Re Py2ðf þ Df ; tþ DtÞ

� �Re Py

1ðf ; tÞ� �� �

¼ 1

2Re CPy

2Py

1ðDf ;Dt; Lz; LyÞ

n o; (17)

where CPz2Pz

1ðDf ;Dt; Lz; LyÞ and CPy

2Py

1ðDf ;Dt; Lz; LyÞ are

given as Eqs. (8) and (9), respectively.

IV. ZCRS OF PARTICLE VELOCITY CHANNELS

Let a(t) be a real random process with the temporal

autocorrelation Ca (Dt)¼E[a(t þ Dt) a(t)]. Then the ZCR of

a(t), the average number of times that a(t) crosses the thresh-

old zero per unit time, can be calculated using the following

formula:35,36

nta ¼

1

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�C00aðDtÞ

��Dt¼0

Cað0Þ

s; (18)

where double prime is the second derivative. Similarly, if

a(f) is a real random process in the frequency domain, then

its ZCR, nfa , the average number of times that a(f) crosses

the threshold zero per unit frequency, is given by

nfa ¼

1

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�C00aðDf Þ

��Df¼0

Cað0Þ

s; (19)

where Ca (Df)¼E[a(f þ Df) a(f)] is the frequency autocorre-

lation of a(f).To study the delay and Doppler spreads of particle ve-

locity channels, we need to calculate frequency- and time-

domain ZCRs of the real parts of the complex channels

Pz(f, t) and Py(f, t). To do this, first the autocorrelation of the

real part of the pressure channel, Re{P(f, t)}, should be

determined. Using Eqs. (5) and (11) we have

CRe Pf gðDf ;DtÞ¼E Re Pðf þ Df ; tþ DtÞf gRe Pðf ; tÞf g½ �

¼ 1

2Re CPPðDf ;DtÞf g: (20)

The autocorrelation of the real part of the vertical particle

velocity channel, Re{Pz(f, t)}, can be written using Eqs. (12)

and (16),

CRe Pzf gðDf ;DtÞ¼E Re Pzðf þ Df ; tþ DtÞf gRe Pzðf ; tÞf g½ �

¼ 1

2Re CPzPzðDf ;DtÞf g: (21)

Similarly, based on Eqs. (13) and (17), the autocorrelation of

the real part of the horizontal velocity channel, Re{Py(f, t)},

is given by

CRe Pyf gðDf ;DtÞ¼E Re Pyðf þ Df ; tþ DtÞf gRe Pyðf ; tÞf g½ �

¼ 1

2Re CPyPyðDf ;DtÞf g: (22)

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A. Frequency-domain ZCRs

Here we have Dt¼ 0. By inserting Eq. (11) into (20),

taking derivative with respect to Df twice and then replacing

Df with zero, for the pressure channel we obtain,

�C00RefPgðDf ; 0Þ���Df¼0¼Kbð2pTbÞ2

2Ecb

1

sin2ðcbÞ

�

þð1� KbÞð2pTsÞ2

2Ecs

1

sin2ðcsÞ

� :

(23)

Note that Ecb ½�� and Ecs ½�� are expected values with respect

to cb and cs, respectively. Similarly, by inserting Eq. (12)

into (21) [or Eq. (13) into (22)], differentiation with respect

to Df twice and then replacing Df with zero, for the vertical

(or the horizontal) velocity channel we obtain,

�C00RefPzgðDf ; 0Þ���Df¼0¼ Kbð2pTbÞ2

2þ ð1� KbÞð2pTsÞ2

2;

(24)

�C00RefPygðDf ; 0Þ���Df¼0¼Kbð2pTbÞ2

2Ecb

cos2ðcbÞsin2ðcbÞ

�

þ ð1� KbÞð2pTsÞ2

2Ecs

cos2ðcsÞsin2ðcsÞ

� :

(25)

Moreover, by inserting Eqs. (11)–(13) into Eqs. (20)–(22),

respectively, it is easy to verify,

CRefPgð0; 0Þ ¼1

2; (26)

CRefPzgð0; 0Þ¼Kb

2Ecb sin2ðcbÞ� �

þð1� KbÞ2

Ecs sin2ðcsÞ� �

;

(27)

CRefPygð0;0Þ¼Kb

2Ecb cos2ðcbÞ� �

þð1�KbÞ2

Ecs cos2ðcsÞ� �

:

(28)

To obtain nfRefPg , nf

RefPzg, and nfRefPyg , frequency-domain

ZCRs, one needs to simply divide Eqs. (23)–(25) by Eqs.

(26)–(28), respectively.

B. Time-domain ZCRs

Now we have Df¼ 0. Similarly to the previous fre-

quency-domain derivations, we can obtain the following

results in time domain,

�C00RefPgð0;DtÞ���Dt¼0¼ Kbð2pfMÞ2

2Ecb cos2ðcb � uÞ� �

þ ð1� KbÞð2pfMÞ2

2Ecs

� cos2ðcs � uÞ� �

; (29)

�C00RefPzgð0;DtÞ���Dt¼0¼ Kbð2pfMÞ2

2

� Ecb sin2ðcbÞ cos2ðcb � uÞ� �

þ ð1� KbÞð2pfMÞ2

2

� Ecs sin2ðcsÞ cos2ðcs � uÞ� �

;

(30)

�C00RefPygð0;DtÞ���Dt¼0¼ Kbð2pfMÞ2

2

� Ecb cos2ðcbÞ cos2ðcb � uÞ� �

þ ð1� KbÞð2pfMÞ2

2

� Ecs cos2ðcsÞ cos2ðcs � uÞ� �

:

(31)

By dividing Eqs. (29)–(31) by Eqs. (26)–(28), respectively,

time-domain ZCRs ntRefPg, nt

RefPzg, and ntRefPyg will be

obtained.

Note that the concept of channel ZCR is not limited to a

certain frequency range. As long as channel correlation func-

tions are available, one can calculate channel ZCRs, as

explained in this section. To represent channels and calculate

their correlation functions, however, we have used a ray-

based framework. So, the derived results are appropriate

for frequencies where ray analysis is relevant, say, high

frequencies.

V. A CASE STUDY

Equations (23)–(31), which determine ZCR and there-

fore delay and Doppler spreads, are valid for any AOA dis-

tribution model. Here we use the Gaussian model,31,37

wbðcbÞ¼ð2pr2bÞ�1=2

exp½�ðcb�lbÞ2.ð2r2

bÞ�;cb 2 ð0;pÞ;

wsðcsÞ¼ð2pr2s Þ�1=2

exp½�ðcs�lsÞ2.ð2r2

s Þ�;cs 2 ðp;2pÞ:

(32)

Note that lb and ls are the mean AOAs from the bottom and

surface, respectively, whereas rb and rs represent angle

spreads. Gaussian PDFs in Eq. (32) are particularly useful

when angle spreads are small.38 Based on experimental

data, Gaussian PDFs with small variances are observed to be

suitable for statistical modeling of an underwater acoustic

channel.37 For large angle spreads, one can use the von

Mises PDF.39

For Gaussian AOA PDFs with small angle spreads,

mathematical expectations in Eqs. (23)–(31) can be calcu-

lated in closed-forms, using the Taylor series expansions of

cb and cs around lb and ls, respectively. For example,

1

sin2ðcbÞ csc2ðlbÞ � 2 csc2ðlbÞ cotðlbÞðcb � lbÞ

þ cosð2lbÞ þ 2ð Þ csc4ðlbÞðcb � lbÞ2; (33)

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where csc(�)¼ 1/sin(�) and cot(�)¼ cos(�)/sin(�). Since

Ecb ½cb�lb�¼0 and Ecb ½ðcb�lbÞ2�¼r2b , Eq. (33) simplifies to

Ecb

1

sin2ðcbÞ

� csc2ðlbÞ þ cosð2lbÞ þ 2ð Þ csc4ðlbÞr2

b

(34)

Following the same approach, one can calculate other Ecb �½ �in Eqs. (25)–(31) as

Ecb

cos2ðcbÞsin2ðcbÞ

� cot2ðlbÞ þ cosð2lbÞ þ 2ð Þ csc4ðlbÞr2

b;

(35)

Ecb sin2ðcbÞ� �

sin2ðlbÞ þ cosð2lbÞr2b; (36)

Ecb cos2ðcbÞ� �

cos2ðlbÞ � cosð2lbÞr2b; (37)

Ecb cos2ðcb � uÞ� �

cos2ðlb � uÞ þ cosð2lb � 2uÞr2b;

(38)

Ecb sin2ðcbÞ cos2ðcb � uÞ� � cos2ðlb � uÞ sin2ðlbÞ þ 0:5 cosð2lbÞðþ2 cosð4lb � 2uÞ � cosð2lb � 2uÞÞr2

b; (39)

Ecb cos2ðcbÞ cos2ðcb � uÞ� � cos2ðlbÞ cos2ðlb � uÞ � 0:5 cosð2lbÞðþ 2 cosð4lb � 2uÞ þ cosð2lb � 2uÞÞr2

b: (40)

Clearly, similar results can be obtained for cs, in terms of ls

and rs.

To obtain some insight, we consider a case where the

bottom components are dominant, i.e., Kb¼ 1. Then we pro-

vide closed-form frequency- and time-domain ZCRs in the

following subsections.

A. Frequency-domain ZCRs

By inserting Eqs. (23)–(28) into the ZCR formula in Eq.

(19) and using the small angle spread approximation in Eqs.

(34)–(37), one can show that

nfRefPg ¼ 2Tb csc2ðlbÞ þ cosð2lbÞ þ 2ð Þ csc4ðlbÞr2

b

� �1=2;

(41)

nfRefPzg ¼ 2Tb

1

sin2ðlbÞ þ cosð2lbÞr2b

" #1=2

; (42)

nfRefPyg ¼ 2Tb

cot2ðlbÞ þ cosð2lbÞ þ 2ð Þ csc4ðlbÞr2b

cos2ðlbÞ � cosð2lbÞr2b

� 1=2

:

(43)

Equations (41)–(43), normalized by Tb, are plotted in Fig. 4.

Simulation results are also included, which show the accu-

racy of derived formulas. In simulations, the depth of the

shallow water channel is 100 m. Center frequency is 12 kHz,

sound speed is considered to be 1500 m/s, number of

received rays is 50, and the speed of the vector sensor re-

ceiver initially located at a 50 m depth is 2.5 m/s.

FIG. 4. (Color online) Frequency-domain ZCRs of particle velocity and

pressure channels versus the angle spread rb(lb¼ 5).

FIG. 5. (Color online) Normalized impulse responses of particle velocity

channel and pressure channel.

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Figure 4 shows the dependence of frequency-domain

ZCRs on the angle spread. As expected, the ZCRs of y-veloc-

ity and pressure channels are very close. This is because the

waves are coming through an almost horizontal direction

(lb¼ 5 ). Increase of ZCRs with the angle spread can be

related to the fact that as the angle spread increases, more rays

from different directions reach the receiver. This means larger

delay spreads, which results in large ZCRs in the frequency

domain. Since in the case considered in Fig. 4, most of the

AOAs are horizontal, the coming rays do not contribute much

to the z-velocity channel. This explains the lower values of the

frequency-domain z-velocity ZCR. This can be better under-

stood by comparing the impulse responses of particle velocity

and pressure channels in Fig. 5. This figure is obtained by

plotting the impulse response in Eq. (6)31 and its derivatives

with respect to y and z, for lb¼ 5 and rb¼ 3. Clearly the

impulse response of the z channel is spread over a small range

of delays in this example. These are consistent with other

delay spread results,34 obtained via a different approach. The

advantage of the proposed velocity channel delay spread anal-

ysis here via frequency-domain ZCR is that it provides analyt-

ical expressions such as Eqs. (42) and (43). These types of

expressions quantify the delay spreads of acoustic particle

velocity and pressure channels in terms of key channel param-

eters such as mean AOAs and angle spreads. Further results

are provided in Appendix B, which shed some light on the

behavior of delay spread in particle velocity channels.

B. Time-domain ZCRs

By inserting Eqs. (29)–(31) and Eqs. (26)–(28) into the

ZCR formula in Eq. (18) and using the small angle spread

approximation in Eqs. (36)–(40), the following results can

be obtained.

ntRefPg ¼ 2fM cos2ðlb � uÞ þ cosð2lb � 2uÞr2

b

� �1=2; (44)

ntRefPzg ¼ 2fM

cos2ðlb � uÞ sin2ðlbÞ þ 12ðcosð2lbÞ þ 2 cosð4lb � 2uÞ � cosð2lb � 2uÞÞr2

b

sin2ðlbÞ þ cosð2lbÞr2b

" #1=2

; (45)

ntRefPyg ¼ 2fM

cos2ðlbÞ cos2ðlb � uÞ � 12ðcosð2lbÞ þ 2 cosð4lb � 2uÞ þ cosð2lb � 2uÞÞr2

b

cos2ðlbÞ � cosð2lbÞr2b

� 1=2

: (46)

Equations (44)–(46), normalized by fM are plotted in Fig. 6,

along with simulation results, which demonstrate the accu-

racy of the analytical results. According to this figure, time-

domain ZCRs of particle velocity and pressure channels are

about the same and not dependent much on the angle spread,

for the case considered. The analysis conducted in Appendix

B for other conditions provides more insight on Doppler

spread in these communication channels.

VI. CONCLUSION

In this paper, a zero-crossing framework is developed to

study the delay and Doppler spreads in multipath underwater

acoustic particle velocity and pressure channels. A geometri-

cal–statistical representation of the shallow water waveguide

is considered, to derive the required frequency- and time-do-

main channel correlation functions. Then the delay and

Doppler spreads are calculated by deriving closed-form

expressions for the channel ZCRs in frequency and time,

respectively. These expressions show how velocity channel

delay and Doppler spreads may depend on some key param-

eters of the channel such as mean AOA and angle spread.

The results are useful for design and performance prediction

of vector sensor systems that operate in acoustic particle ve-

locity channels.

ACKNOWLEDGMENTS

This work is supported in part by the National Science

Foundation (NSF), Grant CCF-0830190.

FIG. 6. (Color online) Time-domain ZCRs of particle velocity and pressure

channels versus the angle spread rb(lb¼ 5, u¼ 0).

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APPENDIX A: PARAMETERS OF THE PRESSURECHANNEL CORRELATION FUNCTION

Here we show how all the parameter and variables in

the pressure channel correlation function in Eq. (5) can be

expressed in terms of cb and cs, the bottom and surface

AOAs, respectively, and also Lz, Ly, D, and D0. According

to Fig. 3, it is straightforward to verify,

sinðcb1Þ ¼ ðD0 � Dþ ðLz=2ÞÞ=nb

1;

sinðcb2Þ ¼ ðD0 � D� ðLz=2ÞÞ=nb

2; (A1)

sinðcs1Þ ¼ �ðD� ðLz=2ÞÞ=ns

1;

sinðcs2Þ ¼ �ðDþ ðLz=2ÞÞ=ns

2; (A2)

cosðcb1Þ ¼ ½ðD0 � DÞ cotðcbÞ � ðLy=2Þ�=nb

1;

cosðcb2Þ ¼ ½ðD0 � DÞ cotðcbÞ þ ðLy=2Þ�=nb

2; (A3)

cosðcs1Þ ¼ �½D cotðcsÞ þ ðLy=2Þ�=ns

1;

cosðcs2Þ ¼ �½D cotðcsÞ � ðLy=2Þ�=ns

2; (A4)

where cot(�)¼ cos(�)=sin(�). Moreover, nb1, nb

2, ns1, and ns

2 are

rays travel distances from the sea bottom and surface to Rx1

and Rx2, respectively, and can be expressed as

nb1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðD0 � DÞ2 þ L2 sin2ðcbÞ � 2ðD0 � DÞL cos

p2þ cb � arctan

Ly

Lz

� �sinðcbÞ

s ,sinðcbÞ; (A5)

nb2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðD0 � DÞ2 þ L2 sin2ðcbÞ � 2ðD0 � DÞL cos

p2� cb þ arctan

Ly

Lz

� �sinðcbÞ

s ,sinðcbÞ; (A6)

ns1 ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2 þ L2 sin2ðcsÞ � 2DL cos

p2þ cs � arctan

Ly

Lz

��sinðcsÞ

s �sinðcsÞ; (A7)

ns2 ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2 þ L2 sin2ðcsÞ � 2DL cos

p2� cs þ arctan

Ly

Lz

��sinðcsÞ

s �sinðcsÞ: (A8)

Moreover, sbq and ss

q in Eq. (5), q¼ 1, 2, are the travel times

from bottom and surface scatterers to the Rx1 and Rx2,

respectively, which are given by

sb1 ¼

nb1

c; sb

2 ¼nb

2

c; ss

1 ¼ns

1

c; ss

2 ¼ns

2

c; (A9)

where c is the sound speed.

By inserting Eqs. (A1)–(A9) into Eq. (5), one obtains

the pressure channel frequency-time-space correlation as

two integrals over the bottom and surface AOAs cb and cs,

respectively. For any given AOA PDFs wb(cb) and ws(cs),

the two integrals in Eq. (5) can be computed numerically.

For some special cases of interest such as vertical and hori-

zontal vector sensor arrays discussed below, Eqs. (A1)–(A8)

can be significantly simplified. This allows to calculate the

integrals in Eq. (5) analytically.34

A. Vertical vector sensor array

For Lz�min(D, D0�D), Ly¼ 0 and usingffiffiffiffiffiffiffiffiffiffiffi1þ xp

1þ ðx=2Þ when |x| � 1, distances nb1, nb

2 , ns1, and

ns2 given in Eqs. (A5)–(A8) can be approximated as

nb1 ½ðD0 � DÞ þ Lz sin2ðcbÞ=2�= sinðcbÞ;

nb2 ½ðD0 � DÞ � Lz sin2ðcbÞ=2�= sinðcbÞ;

ns1 �½D� ðLz sin2ðcsÞ=2Þ�= sinðcsÞ;

ns2 �½Dþ ðLz sin2ðcsÞ=2Þ�= sinðcsÞ: (A10)

Then substitution of Eq. (A10) into Eqs. (A1)–(A4) and

(A9), results in

sinðcb2Þ � sinðcb

1Þ Lz sin3ðcbÞ=ðD0 � DÞ;sinðcs

2Þ � sinðcs1Þ �Lz sin3ðcsÞ=D; (A11)

sinðcb2Þ þ sinðcb

1Þ 2 sinðcbÞ;sinðcs

2Þ þ sinðcs1Þ 2 sinðcsÞ; (A12)

sb1 � sb

2 Lz sinðcbÞ=c;

ss1 � ss

2 Lz sinðcsÞ=c; (A13)

sb2 ðD0 � DÞ=ðc sinðcbÞÞ;

ss2 �D=ðc sinðcsÞÞ: (A14)

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B. Horizontal vector sensor array

Similarly, when Ly � min(D, D0 � D) and Lz = 0, the

distances nb1, nb

2, ns1, and ns

2 given in Eqs. (A5)–(A8) can be

similarly approximated by

nb1 ½ðD0 � DÞ � ðLy sinðcbÞ cosðcbÞÞ=2�= sinðcbÞ;

nb2 ½ðD0 � DÞ þ ðLy sinðcbÞ cosðcbÞÞ=2�= sinðcbÞ;

ns1 �½Dþ ðLy sinðcsÞ cosðcsÞÞ=2�= sinðcsÞ;

ns2 �½D� ðLy sinðcsÞ cosðcsÞÞ=2�= sinðcsÞ: (A15)

Substitution of Eq. (A5) into Eqs. (A1)–(A4) results in

sinðcb2Þ � sinðcb

1Þ �Ly

D0 � D

cosðcbÞ � cosð3cbÞ4

�;

sinðcs2Þ � sinðcs

1Þ Ly

D

cosðcsÞ � cosð3csÞ4

�; (A16)

cosðcb2Þ þ cosðcb

1Þ 2 cosðcbÞ;

cosðcs2Þ þ cosðcs

1Þ 2 cosðcsÞ; (A17)

sb1 � sb

2 �Ly cosðcbÞ=c;

ss1 � ss

2 �Ly cosðcsÞ=c; (A18)

sb2 ðD0 � DÞ=ðc sinðcbÞÞ;

ss2 �D=ðc sinðcsÞÞ: (A19)

Using Eqs. (A10)–(A19) and for Gaussian AOA PDFs

with small angle spreads, on can solve the integrals in Eq.

(5) in closed forms.34 Differentiation with respect to Lz or Ly

provides integral-free expressions for z- and y-particle veloc-

ity channels, respectively.

APPENDIX B: COMPARISON OF VELOCITY CHANNELZCRS

To better understand delay and Doppler spreads of

acoustic particle velocity channels, here we consider the

practical case where most of the rays in shallow water come

along the horizontal direction. This implies that mean AOAs

are relatively small, which enable us to further analyze ve-

locity channel ZCRs. Similarly to Sec. V, we consider the

case where the bottom rays are dominant. With pressure

channel ZCR as a reference, in what follows we compare the

ZCRs of particle velocity channels.

A. Frequency-domain ZCRs

Without less of generality and to simplify the notation,

we consider the square root value of ZCRs. Using Eqs. (41)–

(43), the frequency-domain ZCRs of velocity channels with

respect to the pressure channel can be written as

nfRefPzg

nfRefPg

!2

¼ 1=½sin2ðlbÞ þ cosð2lbÞr2b�

csc2ðlbÞ þ ðcosð2lbÞ þ 2Þ csc4ðlbÞr2b

;

(B1)

nfRefPyg

nfRefPg

!2

¼ ½cot2ðlbÞ þ ðcosð2lbÞ þ 2Þ csc4ðlbÞr2b=cos2ðlbÞ � cosð2lbÞr2

b�csc2ðlbÞ þ ðcosð2lbÞ þ 2Þ csc4ðlbÞr2

b

: (B2)

Using the first-order Taylor series expansions sin(lb) lb,

and cos(lb), cos(2lb) 1, when lb is small, it is straightfor-

ward to verify,

nfRefPzg

nfRefPg

!2

1

1þ 4ðrb=lbÞ2 þ 3ðrb=lbÞ4; (B3)

nfRefPyg

nfRefPg

!2

1

1� r2b

: (B4)

According to Eq. (B3), ZCR of the z-velocity channel can be

smaller than the pressure channel, whereas Eq. (B4) shows

the ZCRs of the y-velocity channel and pressure channels

are nearly the same. These are consistent with the observa-

tions made in Sec. V.

B. Time-domain ZCRs

Based on Eqs. (44)–(46), time-domain ZCRs of z- and

y-velocity channels with respect to the pressure channel are

given as

ntRefPzg

ntRefPg

!2

¼ cos2ðlb � uÞ sin2ðlbÞ þ 0:5ðcosð2lbÞ þ 2 cosð4lb � 2uÞ � cosð2lb � 2uÞÞr2b

cos2ðlb � uÞ þ cosð2lb � 2uÞr2b

�sin2ðlbÞ þ cosð2lbÞr2

b

� ; (B5)

ntRefPyg

ntRefPg

!2

¼ cos2ðlbÞ cos2ðlb � uÞ � 0:5ðcosð2lbÞ þ 2 cosð4lb � 2uÞ þ cosð2lb � 2uÞÞr2b

cos2ðlb � uÞ þ cosð2lb � 2uÞr2b

�cos2ðlbÞ � cosð2lbÞr2

b

� : (B6)

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For u¼ 0 and using the first order Taylor series mentioned

previously, Eqs. (B5) and (B6) can be simplified to

ntRefPzg

ntRefPg

!2

1

1þ r2b

; (B7)

ntRefPyg

ntRefPg

!2

1� 2r2b: (B8)

On the other hand, u ¼ p=2 simplifies Eqs. (B5) and (B6) to

ntRefPzg

ntRefPg

!2

1

1� rb=lbð Þ4; (B9)

ntRefPyg

ntRefPg

!2

1þ rb=lbð Þ2

1� rb=lbð Þ2: (B10)

Here we observe that when the receiver moves horizontally

toward the transmitter, ZCRs of z- and y-velocity channels

are almost the same as the pressure channel ZCR, according

to Eqs. (B7) and (B8). When the receiver moves vertically

toward the bottom, there ratios could be somewhat different.

Given the shallow depth of the channel, this holds only over

a short period of time.

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